# Deltoidal hexecontahedron

Deltoidal hexecontahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramm5o3m
Elements
Faces60 kites
Edges60+60
Vertices12+20+30
Vertex figure20 triangles, 30 squares, 12 pentagons
Measures (edge length 1)
Dihedral angle$\arccos\left(-\frac{19+8\sqrt5}{41}\right) ≈ 154.12136°$ Central density1
Related polytopes
DualSmall rhombicosidodecahedron
ConjugateGreat deltoidal hexecontahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

The deltoidal hexecontahedron, also called the strombic hexecontahedron or small lanceal ditriacontahedron, is one of the 13 Catalan solids. It has 60 kites as faces, with 12 order-5, 20 order-3, and 30 order-4 vertices. It is the dual of the uniform small rhombicosidodecahedron.

It can also be obtained as the convex hull of a dodecahedron, an icosahedron, and an icosidodecahedron. If the dodecahedron has unit edge length, the icosahedron's edge length is $\frac{7+\sqrt5}{6} ≈ 1.53934$ and the icosidodecahedron's edge length is $\frac{4-\sqrt5}2 ≈ 0.88197$ .

Each face of this polyhedron is a kite with its longer edges $\frac{7+\sqrt5}{6} ≈ 1.53934$ times the length of its shorter edges. These kites have one angle measuring $\arccos\left(-\frac{5+2\sqrt5}{20}\right) ≈ 118.26868°$ , the opposite angle measuring $\arccos\left(\frac{9\sqrt5-5}{40}\right) ≈ 67.78301°$ , and the other two angles measuring $\arccos\left(\frac{5-2\sqrt5}{10}\right) ≈ 86.97416°$ .

## Related polytopes

The deltoidal hexecontahedron is topologically equivalent to the rhombic hexecontahedron which has golden rhombi for faces instead of kites.